The quantum free particle on spherical and hyperbolic spaces: A curvature dependent approach II

TL;DR

This paper extends the study of quantum free particles to spherical and hyperbolic spaces, deriving curvature-dependent wave equations, solutions, and spectra, revealing how curvature influences quantum behavior.

Contribution

It introduces a curvature-dependent formalism for quantum particles on curved spaces, deriving explicit wave functions and spectra for spherical and hyperbolic geometries.

Findings

Wave functions related to orthogonal polynomials

Discrete energy spectrum in spherical case

K-dependent deformation of Bessel equation

Abstract

This paper is the second part of a study of the quantum free particle on spherical and hyperbolic spaces by making use of a curvature-dependent formalism. Here we study the analogues, on the three-dimensional spherical and hyperbolic spaces, $S_\k^3$ ($\kappa>0$) and $H_\k^3$ ($\kappa<0$), to the standard {\itshape spherical waves} in $E^3$. The curvature $\k$ is considered as a parameter and for any $\k$ we show how the radial Schr\"odinger equation can be transformed into a $\k$-dependent Gauss hypergeometric equation that can be considered as a $\k$-deformation of the (spherical) Bessel equation. The specific properties of the spherical waves in the spherical case are studied with great detail. These have a discrete spectrum and their wave functions, which are related with families of orthogonal polynomials (both $\k$-dependent and $\k$-independent), and are explicitly obtained.